Find the antiderivative of functions by the method of inspection: `sin 2x-4e^(3x)`

Explanation

To find the antiderivative of the function \( \sin(2x) - 4e^{3x} \) using the method of inspection, we will break down the problem into two parts: finding the antiderivative of \( \sin(2x) \) and finding the antiderivative of \( -4e^{3x} \).

### Step 1: Find the antiderivative of \( \sin(2x) \)

1. We know that the derivative of \( -\cos(x) \) is \( \sin(x) \).
2. To find the antiderivative of \( \sin(2x) \), we need to apply the chain rule. The derivative of \( \cos(2x) \) is \( -\sin(2x) \cdot 2 \) (where 2 comes from the derivative of \( 2x \)).
3. Therefore, the antiderivative of \( \sin(2x) \) is \( -\frac{1}{2} \cos(2x) \).

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